Toss the chunks of potato with some oil and salt and roast at 325 for about an hour until tender. Transfer to a bowl.
Turn the oven to broil on high. Toss the broccoli with some oil and salt and broil to brown around the edges, about 5 minutes. transfer to a food processor.
Add the cheese to the food processor and pulse until combined.
Add the egg and half the breadcrumbs to the potato and mash and mix until smooth. If it feels too wet, add more breadcrumbs.
Pour the remaining breadcrumbs onto a plate.
Scoop up some potato and press it flat in your hand. Scoop about 2 tablespoons of the broccoli mix in the center. Wrap the potato around the broccoli, then roll the ball in the breadcrumbs. Set aside and repeat with remaining potato and filling until you run out. Should be able to make about 8-10
Bring oil to 350. Fry in batches of 2 or 3 to brown on all sides. Remove from heat and serve. I had it with hot sauce.